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Questions tagged [interpolation]

Questions on interpolation, the estimation of the value of a function from given input, based on the values of the function at known points. It is necessary because in science and engineering we often need to deal with discrete experimental data.

Interpolation is a useful mathematical and statistical tool used to estimate values between two points on a line or curve.

What Is Interpolation?

Interpolation is the process of deriving a simple function from a set of discrete data points so that the function passes through all the given data points (i.e. reproduces the data points exactly) and can be used to estimate data points in-between the given ones.

To help us remember what it means, we should think of the first part of the word, 'inter,' as meaning 'enter,' which reminds us to look 'inside' the data we originally had.

Applications: This tool, interpolation, is not only useful in statistics, but is also useful in science, business or any time there is a need to predict values that fall within two existing data points. It is also used to simplify complicated functions by sampling data points and interpolating them using a simpler function. In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate, i.e., estimate the value of that function for an intermediate value of the independent variable.

The details, techniques, and precise meaning of interpolation depend heavily on the sub-discipline of mathematics, therefore you are encouraged to use additional subject tags such as , , , or when appropriate.

2029 questions
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Are my results new?

I'm eighteen and sometimes I like doing math on my own when I'm inspired. I would like to know if some of my "discoveries" are new (I don't think so :) ). These are some of the results I found in the last 3 years: Infinite radical converging to…
Kandinskij
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What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?

I am reading a book about computer graphics. It is confusing about the various splines and their algorithms. What is the difference between natural cubic spline, Hermite spline, Bézier spline and B-spline?
user960456
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Is it impossible to perfectly fit a polynomial to a trigonometric function on a closed interval?

On a closed interval (e.g. $[-\pi, \pi]$), $\cos{x}$ has finitely many zeros. Thus I wonder if we could fit a finite degree polynomial $p:\mathbb{R} \to \mathbb{R}$ perfectly to $\cos{x}$ on a closed interval such as $[-\pi, \pi]$. The Taylor series…
jskattt797
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Find formula from values

Is there any "algorithm" or steps to follow to get a formula from a table of values. Example: Using this values: X Result 1 3 2 5 3 7 4 9 I'd like to obtain: Result =…
Diego
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Accurate floating-point linear interpolation

I want to perform a simple linear interpolation between $A$ and $B$ (which are binary floating-point values) using floating-point math with IEEE-754 round-to-nearest-or-even rounding rules, as accurately as possible. Please note that speed is not a…
Pedro Gimeno
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Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}...$

Is there a smooth, preferably analytic function that grows faster than any function in the sequence $e^x, e^{e^x}, e^{e^{e^x}}$? Note: Here the answer is NOT required to be an elementary function, as I already know that otherwise the answer would be…
blademan9999
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Newton's Interpolation Formula: Difference between the forward and the backward formula

I was taught that the forward formula should be used when calculating the value of a point near $x_0$ and the backward one when calculating near $x_n$. However, the interpolation polynomial is unique, so the value should be the same. So is there any…
Rei
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17
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5 answers

Polynomial fitting where polynomial must be monotonically increasing

Given a set of monotonically increasing data points (in 2D), I want to fit a polynomial to the data which is monotonically increasing over the domain of the data. If the highest x value is 100, I don't care what the slope of the polynomial is at…
splicer
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4 answers

Newton form vs. Lagrange form for interpolating polynomials

I'm just wondering, what are the advantages of using either the Newton form of polynomial interpolation or the Lagrange form over the other? It seems to me, that the computational cost of the two are equal, and seeing as the interpolated polynomial…
Phillip Bredahl
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Why is $L^{1} \cap L^{\infty}$ dense is in $L^{p}$?

It is mentioned that using the interpolation inequality $$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$ one can deduce that the space $L^{1} \cap L^{\infty}$ is dense in $L^{p}$. Does anybody knows the trick behind…
user125967
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How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a subset of other types of curves such as Bezier…
Saras
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General method to "naturally interpolate" to a complex map?

Given a region of the complex plane and a map $z \to f(z)$, is there a general way to "naturally interpolate" the point $z$ to $f(z)$ in such a way that the movement follows a "natural" smooth path that doesn't generate unnecessary "kinks" and…
LucasVB
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Spatial Interpolation for Irregular Grid

How would I interpolate to a point P if I have four points around it such that: Q1 = (x1,y1), Q2 = (x2,y2), Q3 = (x3,y3), Q4 = (x4,y4) If the coordinates formed a regular 2D grid I would use a bilinear interpolation, but I don't think I can use it…
Travis
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Recovering a binary function on a lattice by studying its sum along closed paths

I have a binary function $f:\mathbb N^2\rightarrow\{0,1\}$. While I do not known $f$ explicitly, I have a "device" located at the origin $(1,1)$ which can do the following: Given an even number $m$, the device runs over all closed walks of length…
GSofer
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Interpolation and Taylor's Theorem

I just answered a question where I used the fact that a $(n+1)$-times (continuously) differentiable function $f$ interpolated by a $n$th degree polynomial $p(x)$ through the $n+1$ points $x_0,...,x_n$ has error given by…
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